Math Problem Statement
Solve the system of equations: y = 2x + 1 and 2x - y = 3
Solution
The system of equations in the image is:
y = 2x + 1 \\ 2x - y = 3 \end{cases}$$ To solve this, we can use substitution or elimination. I'll use substitution here since the first equation is already solved for $$y$$. ### Step 1: Substitute $$y = 2x + 1$$ into the second equation. $$2x - (2x + 1) = 3$$ ### Step 2: Simplify the equation. $$2x - 2x - 1 = 3$$ $$-1 = 3$$ This equation is a contradiction, which indicates that there is no solution to this system. Therefore, the system is **inconsistent**. ### Conclusion The system has no solution because the lines represented by these equations are parallel and never intersect. Would you like further explanation on this solution, or do you have any specific questions? Here are 5 related questions for deeper exploration: 1. What does it mean for a system of equations to be inconsistent? 2. How can we tell if two lines are parallel just by looking at their equations? 3. What are other methods to solve a system of equations, such as elimination or graphical solutions? 4. How can we determine if a system has a unique solution, no solution, or infinitely many solutions? 5. What happens if we change one of the constants in this system? How does it affect the solution? **Tip:** When solving systems of equations, checking for consistency (whether a solution exists) is crucial, as some systems may have no solution due to parallel lines.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Linear Equations
Substitution Method
Formulas
y = mx + b
Substitute y in the second equation
Theorems
Consistency and inconsistency in systems of equations
Suitable Grade Level
Grades 8-10